How to use the matrix inverse calculator
Start by choosing the matrix dimension. The calculator builds a square grid, joins the matrix to an identity matrix, and applies Gauss-Jordan elimination. When the left side becomes the identity matrix, the right side is A^-1. The entries stay as reduced exact fractions, and the row operations remain visible.
- Set
Size (n × n)to the matrix dimension. - Fill every matrix cell with a number or fraction.
- Click
Calculateto parse the entries and reduce [A | I]. - Read
Matrix size,Rank, andStatus. UseCopy resultorDownload result (.txt)for a text copy.
Calculate the inverse of a square matrix
Enter the two-by-two matrix below to see how exact fractions appear in the result. The calculator keeps the fractional entries instead of turning them into rounded decimals.
Input: [1 2]
[3 4]
Output: [-2 1]
[3/2 -1/2]The step list can show row swaps, pivot normalization, and elimination in a pivot column. Every step includes the current augmented matrix, with the original matrix on the left and the identity or inverse side on the right.
What can I enter in each matrix cell?
Each cell accepts one plain number or fraction. Signed integers such as -3, decimals such as 2.5, and fractions such as 3/4 are valid. A blank cell, invalid character, or zero denominator stops the calculation and identifies the affected cell.
Variables and expressions are rejected. For example, x+1 fails because the parser accepts digits, an optional decimal point, an optional leading sign, and one slash for a fraction.
What the matrix inverse result includes
For an invertible matrix, the result area displays A^-1, the matrix size, the rank, and Invertible as the status. The visible steps show each change to the augmented matrix. Copy result and Download result (.txt) include the input matrix, inverse, and rank, while leaving out the displayed elimination steps.
The Size (n × n) control creates square grids from 1×1 through 8×8. Changing the size rebuilds the grid and hides the current result and error. Reset rebuilds a blank grid and hides those states too.
What happens when a matrix has no inverse?
If elimination produces fewer pivots than the matrix size, the calculator reports Singular - no inverse and shows the rank. It leaves out an inverse. Exact fraction arithmetic lets the calculation test each pivot as zero or nonzero.
Where a matrix inverse comes up
For a worked exercise, enter the matrix and compare the final inverse with the handwritten row operations. The rank and singular status also identify a matrix that cannot be inverted.
When checking a small matrix routine, enter the same values, inspect each reduction step, and copy the final matrix into a text note. The entries are unitless numbers; the calculator does not attach physical units.
Which matrix calculator should I use?
Use this calculator for A^-1 and its row-reduction trail. Choose the Matrix Calculator for matrix addition, subtraction, multiplication, transpose, determinant, rank, or a solution to Ax = b. Choose the Matrix Determinant Calculator when you need a square matrix's determinant. For equations, geometry, sequences, vectors, or other number work, start with Math Calculators.
Frequently asked questions
How does the matrix inverse calculator find A^-1?
It row-reduces the augmented matrix [A | I] with Gauss-Jordan elimination. When the left side becomes the identity matrix, the right side is A^-1.
How do I calculate the inverse of a 2×2 matrix?
Set Size (n × n) to 2, fill the four cells, and click Calculate. The result shows the inverse for an invertible matrix, its rank, and the reduction steps.
Why does my matrix have no inverse?
Your matrix is singular when the calculator finds fewer pivots than its size. It reports Singular - no inverse and shows the rank instead of producing an inverse.
Can I enter fractions in the matrix inverse calculator?
Yes, enter a fraction such as 3/4 in a cell. The parser also accepts signed integers and decimals. Blank cells, expressions, invalid characters, and zero denominators produce an error.